I think he spend the entire series looking for the Doctor. I'm not sure; haven't watched it since they killed Owen and Tosh. It was also cool to discover that every character in Torchwood has a cameo role in Doctor Who.

Great show, I just never understood why literally all of the main characters are bisexual
It makes sense for Jack, because he grew up in the somethingith century, where it's the societal norm
It doesn't make sense for everyone else, because they all come from now. It also makes it confusing to watch at times, because you're never exactly sure who wants to **** whom.